Greater Aberrants

Introductions

Greater Aberrants are one of the most terrifying - and dangerous - enemies that one can encounter out there. They are powerful and nigh-completely incomprehensible magical entities that are extremely hard to even categorize or describe, much less fight.
As a result, dealing with them is one of the main jobs of national agencies for supernatural defenses, similarly to Disaster-class gods.

Nature

Greater Aberrants are, basically speaking, more unique aberrants. Whether they are connected to Powers in a similar way to them is, actually, only one of many hypotheses. What is certain is that each of them are completely unique (and always hostile) entities, with their own strengths and weaknesses, As a result, proper reconaissance is crucial in defeating them. Failing that will inevitably lead to massive casualties.
It is unknown where, when and how exactly the greater aberrants are born. It's most certainly not happening on Earth, as their very biology is naturally incompatible with this realm. They are speculated by some to be coming from either certain Elsewheres (which they certainly find much more agreeable), which is almost certainly not true - most of them are simply way too alien.
As a result, it is generally speculated that at least higher category greater aberrants are hailing from unspecified places outside of reality as known to men. This, in fact, led to many speculation as to whether they actually count as aberrants. They share some traits, and some speculate them to be more conscious creations of Powers (rather than lesser aberrants who are speculated to merely be the equivalent of Power's thoughts).
In the end, though, it might simply be that the world of the initiates is too enlightened to follow up on their counterparts being gods and simply call them demons.

Taxonomy

Greater Aberrants are typically separated not accordingly to their power level (as that's extremely hard to quantify in many cases) but accordingly to how much they continue to adhere to the rules of the physical world. This is extremely important as those levels are not only more or less distinct, but they are also crucial to any subjugation plan due to the way a higher category aberrants influence people around them.
Category One - Greater Aberrants that, in fact, are barely worth being referred to as greater aberrants. They are almost entirely composed of conceivable concepts. While they always get a lot of things wrong, you'll at the very least recognize what they were supposed to be. This means that they are creepy rather than inconceivable, leading to practical absence of sanity and corporeal damage. They are capable of freely operating in the real world, thanks to that. Those entities are also incapable of warping space on their own.
Category Two - This is where things are starting to become complicated. Cat Two greater aberrants are still at 'fundamentally wrong' stage rather than downright mindbreaking, but there'll be a recognizable damage to sanity of everyone in the vicinity, even if it's still mixed with creepiness (and the latter triumphs). They CAN operate in material reality and will do it, but except them to - unlike the Cat One - to begin to warp reality around them to fit them slightly better.
Category Three - That's where things go from complicated to downright troubling. Cat Three is past being a mixture of wrong and conceivable. There are still more or less recognizable, but rather than a description you can expect a rough vibe - say 'plant' or 'animal'. This is still enough for human brain to latch to at least something, downscaling the damage to sanity and corporeal stability. At this point Greater Aberrants begin to have issues showing up in person in real world, and will instead (if it even show up there) hide in its own foldspace, requiring an Invocation Glyph to be dragged out to fight it.
Category Four - This is basically where all matter of logic flies out of the window. They'll still have at least some recognizable shape and form, although it will still be incredibly hard to even look at them, as even that act will cause progressive crumbling of one's sanity and corporeal stability. They are also the highest category that tends to enter Reality of their own volition, but it'll be almost always a short stay as the reality itself is highly unpleasant to them. They can be encountered in the Elsewheres, where they tend to act as Cat Three would on Earth.
Category Five - Now we are beginning to enter the field where any degree of comprehension is basically impossible to be described properly, although it can still be at least somewhat imagined. In a way, it's the border where there is still some comprehension left, but human linguistic skills end up running dry. This is where some of the more logical Elsewheres begin to be perceived as still way too simple.
Category Six - This is the big nope moment. Cat Six is best described as the most divergent greater aberrant category that can still spend extended periods of time in any Elsewhere where humans can still be at least technically seen. We talk about lower levels of Abyss and similarly "nice" locations. They can still exist and dwell there, but in most cases they'll still hide in a foldspace to make it nicer. They won't look like anything you can comprehend, but at the very least, you can look at them and at least try.
Category Seven - Yeah, nope. Category Seven is an umbrella term for everything past Category Six. Those entities are extremely, extremely rare. Just as other greater aberrants sometimes show up where they aren't supposed to, Cat Seven entities sometimes show up at all (which is already something they weren't supposed to). They are described as not just being too alien to comprehend and thus causing sanity damage, but in fact so alien that even perceiving them will snap your mind like a twig.
The rule of thumb is that Category One to Category Three can be engaged by sufficiently hardened mages, cultists, demihumans and Contract or weaker Authority Stage warlocks, although against Category Three you'll need to be quick or you'll suffer lasting damage quite quickly. Category Four to Category Six requires involvement of magisters, heresiarchs, nightmares or stronger Authority/Terminal Stage warlocks.
Category Seven ones aren't supposed to be fought. Instead, you're supposed to try to run away from them, only to promptly fail and die a horrible death (or, worst of all, don't die and only wish you did). The contingency plan for most of the major national agencies in case of one of them wandering into Earth is to drop a nuke on it and pray to any gods out there that it'll inconvenience them enough to leave.

Additional Notes

1. Greater Aberrants are especially dangerous for one reason - they'll actively hunt non-initiated. This isn't the same case as with other supernatural entities - lesser aberrants tend to be drawn towards the initiates, undead tend to hit people that were to various degree connected to them, gods will typically ignore them (though accidental collateral damage is a possibility) and demihumans will typically do whatever they want. Greater aberrants will, however, attack all humans in the vicinity. Initiates or not.
2. In certain situations, point one aside, they can be beneficial. They draw magic to themselves, resulting in greater ease of access to certain paramaterials and alchemical components. For example, a foldspaced plant-themed Category Three will cause magical plants to grow in its vicinity. However, tolerating such an entity will always be a rather dangerous game, and requires balancing the threat the greater aberrant is with the benefits that come from its existence.
3. Greater Aberrants that aren't properly researched are utterly unpredictable. You might encounter a greater aberrant that's made from what appears to be wood that will turn out to be completely and perfectly fireproof (or even grow stronger when hit by fire magic), because their connection to conventional logic is very surface level. Unless you have cold hard data about them, assume everything will kill you. Take that into account.

4. Each Greater Aberrant encountered by Mankind receives a name. You can't expect Greater Aberrants to be in any way human enough to use names. In many cases you can't even expect them to know what names are. This has became pretty much a worldwide tradition, tied to many old superstitions implying that the very act of naming them makes a bit more conceivable.
Those names will typically refer as much of their nature as possible to understood, as a way of further anchoring their existence into conceivable forms. The list of encountered greater aberrants (including their names, description and present status) is at least theoretically supposed to be reported to the UN Supernatural Security Committee, however this also includes the members of it, and even that depends on the mood of the people in question.
5. It's hard to properly quantify how strong they are. The easiest comparison between them and their more or less counterpart gods can be summarized as 'more or less comparable'. Greater Aberrants have an advantage of numbers, unpredictability of their nature and sanity/corporeal damage. Gods, however, are typically more powerful in terms of sheer firepower, and typically by far (unless their domains are extremely incompatible). They still need to be on their guard when engaging, or things might go bad for them.

Examples

Serpentine Hound of the Black Forest - A Category One greater aberrant occassionally encountered in a forest close to one of the American magical schools. It appears outwardly similar to a large, black wolf-like creature... at a distance. When you get closer, it'll stand up - and then you'll discover that it has a flexible neck and a mouth opening like the one of a snake, except filled with sharp teeth resembling that of a wolf. Before it cleanly snaps your head off.
Tree that Dreams - A Category Three greater aberrant of plant-like nature. For reasons unknown it has entered Earth through the dreams of a patient in a mental hospital and promptly enveloped the whole places in its numerous roots, while populating it with the warped remnants of humans that were there at the time, remade in its image. It has then retreated into its own foldspace, and remained there ever since.
Dragon from the Desert of Bones - A Category Seven greater aberrant encountered exactly one time, and in an Elsewhere very close to Earth, a clear anomaly suggesting that it was bound there by something greater. It's description refers to its role (that of a dragon set there to guard a 'princess') and location (a desert of grind-up human bones).